Photonic crystals: a medium exhibiting anomalous cherenkov radiation

ABSTRACT

A system for exhibiting Cherenkov radiation is provided. The system includes a beam of charged particles. A photonic crystal structure receives said beam of charged particles. The charged particles moves in said photonic crystal structure so that Cherenkov radiation is produced at all velocities without requiring resonances in the effective material constants of said photonic crystal structure.

PRIORITY INFORMATION

This application claims priority from provisional application Ser. No.60/413,799 filed Sep. 26, 2002, which is incorporated herein byreference in its entirety.

This invention was made with government support under Grant NumberDMR-9808941 awarded by the NSF. The government has certain rights in theinvention.

BACKGROUND OF THE INVENTION

The invention relates to the field of Cherenkov radiation, and inparticular to using photonic crystals as a medium to exhibit anomalousCherenkov radiation.

Cherenkov radiation (CR) is the coherent electromagnetic response of amedium driven by the swift passage of a charged particle. It is thus aneffect strongly dependent on the medium dispersion. In a uniform,isotropic medium with frequency-independent permittivity ∈ andpermeability μ, the condition for CR is well-known, where the velocityof the particle v must exceed the phase velocity of the mediumυ_(ph)=c/√{square root over (∈μ)}. For a dispersive medium, such as anonmagnetic material with a Lorentz-form dielectric response ∈(ω),υ_(ph)(ω)=c/√{square root over (∈(ω))}, is a function of frequency ω.Because ∈(ω) can reach arbitrarily high values near a resonance, it waslong recognized that CR in a dispersive medium can happen for smallcharge velocities, e.g., υ<υ_(ph)(0). The sub-threshold CR in a materialnear its phonon-polariton resonance was disclosed in Cherenkov Radiationat Speeds Below the Light Threshold: Phonon-Assisted Phase Matching, byT. E. Stevens, J. K. Wahlstrand, J. Kuhl, and R. Merlin, SCIENCE, vol.291, No. 5504 (26 Jan. 2001).

On the other hand, it was conjectured that there could exist anotherclass of materials which have both ∈ and μ being negative, henceforthreferred to as “negative index materials.” The properties of suchmaterials were disclosed in The Electrodynamics of Substances withSimultaneously Negative Values of ∈ and μ, by V. G. Veselago, SOVIETPHYSICS USPEKHI, vol. 10, No. 4 (January-February 1968). It wassuggested that a negative index material would reverse many of thewell-known laws of optics. In particular, CR effect is predicted to bereversed, i.e., a fast-moving charge in a negative index medium shouldradiate in the direction opposite to that of its velocity.

A further possibility exists when the charged particle travels near aperiodic structure, where simple Bragg scattering can give rise toradiation without any velocity threshold. This phenomenon (theSmith-Purcell effect) was disclosed in Visible Light from LocalizedSurface Charges Moving across a Grating, by S. J. Smith and P. M.Purcell, PHYSICAL REVIEW, vol. 92, No. 4 (15 Nov. 1953). The radiationdue to traveling charged particles has since been studied inone-dimensionally periodic multilayer stacks in Cerenkov Radiation inInhomogeneous Periodic Media, by K. F. Casey, C. Yeh, and Z. A.Kaprielian, PHYSICAL REVIEW, vol. 140, No. 3B (8 Nov. 1965), and nearthe surface of dielectric structures in Interactions of Radiation andFast Electrons with Clusters of Dielectrics: A Multiple ScatteringApproach, by F. J. Garcia de Abajo, PHYSICAL REVIEW LETTERS, vol. 82,No. 13 (29 Mar. 1999).

SUMMARY OF THE INVENTION

According to one aspect of the invention, there is provided a system forexhibiting Cherenkov radiation. The system includes a beam of travelingcharged particles. A photonic crystal structure receives the beam ofcharged particles. The charged particles move in the photonic crystalstructure so that CR is produced at all velocities without requiringresonances in the effective material constants of said photonic crystalstructure.

According to another aspect of the invention, there is provided a methodof exhibiting Cherenkov radiation. The method includes providing a beamof traveling charged particles. Also, the method includes providing aphotonic crystal structure that receives the beam of charged particles.The charged particles move in the photonic crystal structure so thatCherenkov radiation is produced at all velocities without requiringresonances in the effective material constants of said photonic crystalstructure.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of a radiation mode in a photonic crystal;

FIG. 2 is a schematic diagram of a radiation cone;

FIG. 3A is a TE bandstructure of a 2D square lattice with air columns;FIG. 3B is schematic diagram illustrating a moving charge in thephotonic crystal;

FIGS. 4A-4C are graphs illustrating the relationship between thefrequency, the wavevector k, and the corresponding group velocity u ofthe radiation modes in the first band of the photonic crystal of FIG.3B;

FIGS. 5A-5D are schematic diagrams illustrating the results of theradiation of a charge moving in the photonic crystal of FIG. 3B withv=0.1 c, 0.15 c, 0.30 c, and 0.6 c;

FIGS. 6A-6D are graphs showing the distribution of the radiationmagnetic field perpendicular to the 2D plane with v=0.1 c, 0.15 c, 0.30c, and 0.6 c;

FIGS. 7A-7D are graphs showing the frequency spectrums of the radiationflux along the z-axis with v=0.1 c, 0.15 c, 0.30 c, and 0.6 c;

FIG. 8 is a schematic diagram illustrating a particle detector; and

FIG. 9 is a schematic diagram illustrating a radiation source.

DETAILED DESCRIPTION OF THE INVENTION

The invention describes using photonic crystals, i.e. periodic latticesfor electromagnetic waves, as a medium which exhibits Cherenkovradiation (CR) and proposes new devices that make use of this effect.When a photonic crystal is coherently excited with a beam of travelingcharged particles, the CR is produced. The radiation has its origin inboth the transition radiation, which occurs when the charge experiencesan inhomogeneous dielectric environment provided by the crystal, and theconventional CR in a uniform material, in which coherence is preservedthroughout the medium. Unlike the Smith-Purcell effect, in which lightis generated near a periodic grating and then propagates down through auniform medium, the CR in this invention is generated and propagatesthrough the same crystal in the form of Bloch waves.

Due to the very complex Bragg scattering effect and the rich photondispersion relations in a photonic crystal, the properties of Blochwaves can be very different from waves in a uniform medium, leading to avariety of unusual phenomena. In particular, a charge moving in aphotonic crystal radiates at all velocities without requiring the usualCherenkov threshold condition or resonances in the effective materialconstants. Moreover, the invention predicts new CR wavefront patternsthat are impossible to achieve within either a uniform medium or theSmith-Purcell effect. Furthermore, the invention demonstrates situationsin which CR propagates backward, a behavior reminiscent of thatpredicted in negative-index materials, without requiring the materialconstants to become negative. The invention provides systematic methodsfor analyzing these new phenomena, and confirms predictions by directnumerical simulations.

By discussing the general condition for Cherenkov radiation in photoniccrystals, it is important to consider a particle of charge q moving at aconstant velocity v on a z axis inside a photonic crystal. An analyticalexpression for the fields generated by such a moving charge can bederived using the standard normal-mode expansion in Fourier space as:$\begin{matrix}{E = {\sum\limits_{kng}{\frac{4\quad \pi \quad {\left( {\left( {k + g} \right) \cdot v} \right)}{{qv} \cdot e_{{kn}{({- g})}}^{*}}}{N\quad {\Omega \left( {\omega_{kn}^{2} - \left( {\left( {k + g} \right) \cdot v} \right)^{2}} \right)}}E_{kn}^{{- }\quad {{({k + g})} \cdot {vt}}}}}} & {{Eq}.\quad 1}\end{matrix}$

Here, the wavevector k is summed over all points in the first Brillouinzone, the band index n runs over all bands, and the reciprocal-latticevector g is summed over all the reciprocal space.$E_{kn} = {\sum\limits_{G}{e_{knG}^{\quad {{({K + G})} \cdot r}}}}$

is the eigenmode of the photonic crystal at Bloch wavevector k and bandindex n with corresponding Fourier component e_(knG) and eigen-frequencyω_(kn) (the photonic band structure), and is normalized according to1/Ω∫dr∈(r)E_(kn)E*_(lm)=δ(k−1)δ_(nm) with Ω being the volume of aspatial unit cell of the photonic crystal. N is the number of the unitcells of the crystal, and NΩ is the total volume under consideration.

In the well-studied case of a uniform medium in which simple analyticalexpressions are available for E_(kn) and ω_(kn), Eq. 1 can be shown tobe equivalent to the familiar Fermi results. In the case of a generalphotonic crystal, the far-field radiation modes can be deduced from thepoles in Eq. 1. If restricted to positive frequencies, one can find thecondition for radiation to be ω_(kn)=(k+g)·v. Each solution set of k, nand g then corresponds to a CR mode emitted by the photonic crystal. Therelative excitation strength of each mode is proportional to themagnitude of e_(kn(−g)) multiplied by slow functions of k and g on thenumerator of Eq. 1. Since ω_(kn)=ω_((k+g)n), the radiation condition mayalso be written as

ω_(kn) =k·v  Eq. 2

with k now in an arbitrary Brillouin zone. In this way, g is just thereciprocal-lattice vector required to reduce k to the first Brillouinzone.

Eq. 2 is the general condition for CR in a photonic crystal, and can beeasily reduced to the usual threshold condition mentioned earlier in thecase of a uniform dielectric medium. In a general photonic crystal, itmay be solved by intersecting the plane k·v=ω in k-space with thephotonic-crystal dispersion surface ω_(kn)=ω in the periodic zonescheme, as shown in FIG. 1. The dispersion surface is similar to anormal surface in crystal optics, but is the surface formed by k insteadof the phase velocity. For each fixed n, solutions to Eq. 2 exist forarbitrary v. This is because, as ω goes from the minimum to the maximumfrequency of band n, the dispersion surface traverses through allpossible k-points, and the plane k_(z)=ω/υ only sweeps through a regionwith finite width in k_(z) in k-space. Thus, no threshold exists for CRin photonic crystals. In particular, a moving charge can radiate evenfor υ→0. Physically, this effect is similar to the Umclapp process insolid-state physics. As υ→0 the solution for smallest frequency existsat ω≈g₀·v, g₀ being a primitive reciprocal-lattice vector with smallestpositive g₀·v. Note that since the radiation frequency goes to zero asυ→0, the present effect is fundamentally different from a similar resultin dispersive medium, which requires ∈→∞ and radiation can only be neara fixed resonance frequency. In principle, this applies to any g, butthe coupling strength, which is proportional to |e_(kn(−g))|, must go tozero for large g.

Positive frequency, as described in Eq. 2, requires that k·v>0. In auniform medium with positive index, energy flows along k, and thus amoving charge emit energy forward, i.e., the direction of radiationmakes an acute angle with v. In a negative index material, the situationis reversed. The angle between energy flow and the velocity is nowobstuse, and a moving charge must radiate backwards.

In the case of a general photonic crystal, the direction of energy flowshould be determined from the direction of the group velocity, i.e. thePoyning vector associated with the solution to Eq. 2. In the periodiczone scheme, there are many bands which have group velocities oppositeto that of k. If these bands are coupled to an appropriate v in Eq. 2,then there is a CR in the direction opposite to that of v. Therefore, anegative group-velocity is a sufficient condition for reversed the CReffect. Note that this condition can be applied in the long-wavelengthlimit in the second Brillouin zone, where the photonic crystal exhibitspositive effective index.

Another characteristic of CR is the shape of the radiation wavefront. Ina uniform nondispersive medium, it is well-known that a forward-pointingshock-front is produced on a cone behind the charged particle (thepresence of dispersion removes the singularity). The difference betweengroup velocities and phase velocities also alters the shape of theradiation cone. Here the invention presents a graphical method fordetermining the shape of the cone, as shown in FIG. 2, which can beeasily extended to the case of photonic crystals. FIG. 2 illustrates aplot having both v and the group velocities u of all modes in Eq. 2 in avelocity-space. The trace of u is analogous to the ray surface incrystal optics, though in the present case it is formed by the groupvelocities of different frequencies. The magnitude of u is proportionalto the distance traveled by the wavefront of the corresponding mode, andthe magnitude of v is proportional to distance traveled by the movingcharge. Thus, for each mode, the wavefront of radiation still lies on acone whose angle is determined by the angle θ between v−u and v. Themaximum of all θ's determine the angle of the overall cone thatencompasses all radiation. Propagating Bloch modes can only exist on therear side of this overall cone, whereas the fields are evanescent on theforward side, and across the cone the radiation field amplitudeexperiences a drop.

From these considerations it is understood that the group velocities ofthe modes determined by Eq. 2 are a key to understanding CR in photoniccrystals. They can be calculated either from the gradient vectors to thedispersion surface or from the Heynman-Feynman Theorem specialized tophotonic crystals. The detailed behavior of group velocities in aspecific photonic crystal is studied further hereinafter.

For concreteness, a specific problem in 2D is addressed to illustratethe above analysis. Suppose a charge moves inside an infinite 2D squarelattice of air columns in dielectric ∈=12, as shown in FIG. 3A. Thecolumns have a radius of 0.4 a, where a is the period of the lattice,and the charge is moving in the 2D plane in the (10) lattice direction.The photonic-crystal TE (in-plane electric field E, appropriate for CR)bandstructure and the geometry of this problem are shown in FIG. 3B. Eq.2 is solved to determine the radiation modes of this photonic crystaland their group velocities, using software that calculates the photonicbands by preconditioned conjugate-gradient minimization of the blockRayleigh quotient in a planewave basis. For purposes of discussion, onlythe frequencies in the first band are analyzed and those in higher bandscan be discussed similarly. The results are plotted in FIGS. 4A-4C forseveral different v's.

It is observed that the phase velocity of this photonic crystal in thelong-wavelength limit (ω→0) is υ_(c)=0.44 c. For v<<vc, the radiationcoalesces into resonances around ω≈G·v. For larger v, the resonancesmerge together to form emission bands outside which CR is inhibited(FIG. 4A). As v increases, k and u within each emission band arestrongly influenced by the photonic band structure (FIGS. 4B-4C). Inparticular, for v<υ_(c) most solutions exhibit negative z-components ofgroup velocities, indicating backward-propagating Cherenkov effect forv<υ_(c) in general. Since radiation is allowed even for v<<υ_(c), somemodes can have group velocities that exceed v. According to the analysisof FIG. 2, modes traveling faster than the moving charge produce eithernearly isotropic wavefront in all directions (v≦0.1 c), or reversedradiation cones that point backward (v=0.15 c). While the Smith-Purcellradiation always has a similar isotropic wavefront, it cannot produce areversed radiation cone. The reversed cone effect is also forbidden bycausality in any uniform passive medium, and is therefore a featureunique in photonic crystals.

To confirm these predictions finite-difference time-domain simulationsare performed of the CR in photonic crystals, using a thick boundaryperfectly matched layer (PML) which consists of 10 periods of identicalphotonic crystal buried inside. This boundary condition can effectivelyabsorb radiation modes in photonic crystals away from the band edge. Themoving charge is implemented as a dipole of constant amplitude whichpoints toward the direction of motion z and whose position depends ontime. To verify negative CR, the fields are recorded on all points of anobservation line perpendicular to z and the frequency components of thez-flux are calculated from those of the observed field values.

FIGS. 5A-5D show the calculated group velocities and expected radiationcone shapes. The simulated radiation field pattern and the frequencyspectrum of the z-flux are shown in FIGS. 6A-6D and 7A-7D, respectively.FIGS. 6A-6C demonstrate the radiation below υ_(c), and FIGS. 7A-7Cexhibit the negative z-flux frequency regions which are in agreementfrom what can be obtained from Eq. 2. Moreover, FIG. 6A shows thenear-isotropic radiation at v=0.1 c, and FIG. 6B confirms the reversedCR cone. From FIG. 6, the angular distribution of radiation is alsoreadily available. When v=0.1 c or 0.15 c, the radiation is distributedover a wide range of emission angles without producing a cone ofintensity maxima. For v=0.3 c or 0.6 c, however, the CR becomescollimated, and a definite emission angle in both the forward and thebackward direction for most of the radiation energy can be observed.Note in FIGS. 6D and 7D that the CR radiation at high velocities(v>υ_(c)) in the first band are qualitatively similar to that in auniform medium with very small dispersion (i.e. forward-pointingradiation cone and positive z-flux).

An important difference between radiation patterns for v<υ_(c) andv>υ_(c) is that the DC components (ω=0) extends beyond the radiationcone in FIGS. 6A-6C, while in FIG. 6D the field outside the radiationcone is strictly zero. These DC components are also reflected in thestrong peaks around ω=0 shown in FIGS. 7A-7C. The finite flux values atω≠0 before the onset of negative z-flux is an artifact of finitecomputational size and time, and the DC components should not influenceregions of ω≠0 in ideally finite systems. Finally, it can be noted thatthere are strong, high-frequency “tails” of radiation behind the chargein all cases shown in FIGS. 6A-6D. These correspond to the radiation inthe higher bands which have much smaller group velocity. Of course, herethese tails can exhibit a backward radiation effect as well.

Although a 2D “air-holes-in-dielectric” structure has been previouslyconsidered, the same physics obviously applies to 3D photonic crystalsin general with little change. In particular, dielectric cylinders orspheres in air appear to be a good candidate for experimental studies ofthese effects. The charged particles can be chosen to be electrons,which with high velocities should be available from emission through avoltage difference of several tens or hundreds of kilo-volts. They canthen be directed to travel in the all-air channels of these photoniccrystals. Direct experimental verification of the anomalous CR effectsin photonic crystals introduced in this invention should thus be verypossible.

Particles traveling at speeds below the phase-velocity threshold cannotbe detected by conventional CR counters, and currently their observationrelies on other devices, such as scintillation counters, proportionalcounters, or cloud chambers. These other devices, however, lack theunique advantages of strong velocity sensitivity and good radiationdirectionality as in conventional CR. With a photonic crystal, oneshould be able to achieve velocity selectivity and distinctive radiationpatterns without any velocity threshold. Using the invention, one canform a particle detector 20, as shown in FIG. 8, for counting of chargedparticles can now be made at arbitrary velocities using sensitiveradiation detection devices, such as photomultipliers or vacuumphotocell coupled to an amplifier. Charge particles 26 of all velocitiesare received in the all-air channel of a photonic crystal 28. The CRemitted by the charges is received by the photomultipliers 22surrounding the photonic crystal 28, and the signals 30 can be amplifiedin an external amplifier. Note that waveguides may also be integratedinto the photonic crystal to facilitate radiation collection. Thesensitivity of this detector 20 only relies on the sensitivity of thephotomultipliers 22 and does not require a velocity threshold. A numberof devices measuring particle number, speed, charges or density flux canbe built based on the detected radiation spectrum and wavefront pattern.Such devices are useful wherever particles of arbitrary energy areproduced, such as in monitoring and controlling of nuclear reactors.

Moreover, a radiation source design 36 can be formed using CR, as shownin FIG. 9. An intense electron beam 44 is generated from high-voltagecathode 40 emission and pass through the all-air channel of a photoniccrystal 42. It radiates and the radiation 46 can be collected bywaveguides 48 that are built inside the photonic crystal 42. Of course,the radiation can also travel as bulk Bloch waves as discussed above andbe collected by external devices without using output waveguides. Thefrequency of the radiation depends on its velocity and is tunablethrough the potential applied to the electron beam 44. Electromagneticwave of arbitrary frequencies can now be generated using the effects asshown in FIG. 9. The range of the frequency of radiation 46 is now setby the photonic crystal 42 and the transparent spectral regime of thedielectric, thus selectively scalable beyond the optical wavelengths.Moreover, the frequency is tunable at will by changing the velocity ofthe particles. Furthermore, integrated photonic crystal line-defects canbe employed to concentrate and guide the radiation. A “dense” electronbeam may also lead to coherence in the output radiation. This type ofradiation sources allows very flexible designs, and will be especiallyattractive for frequencies that are otherwise difficult to access, e.g.in terahertz regime.

All conventional applications of CR in the high-velocity regime, such asvelocity measurement and selection, should be able to benefit from thisinvention. Since the particle can be made to travel in the air channel,the disturbance to the particle motion due to impurity scattering andrandom ionization, which introduce inherent losses in a conventionalhigh-index material for CR detectors, is now completely absent. Thiscould greatly enhance the performance of present CR detectors.

Although the present invention has been shown and described with respectto several preferred embodiments thereof, various changes, omissions andadditions to the form and detail thereof, may be made therein, withoutdeparting from the spirit and scope of this invention.

What is claimed is:
 1. A system for exhibiting Cherenkov radiationcomprising: a beam of traveling charged particles; and a photoniccrystal structure that receives said beam of charged particles, saidcharged particles move in said photonic crystal structure so thatCherenkov radiation is produced at all velocities without requiringresonances in the effective material constants of said photonic crystalstructure.
 2. The system of claim 1, wherein said beam of chargedparticles comprise of an electron beam.
 3. The system of claim 2,wherein said photonic crystal structure comprises of output waveguideswhere said Cherenkov radiation outputs said photonic crystal structure.4. The system of claim 3, wherein said Cherenkov radiation is receivedby one or more photomultipliers.
 5. The system of claim 4, wherein saiddispersion of charges particles are absorbed by an absorber.
 6. Thesystem of claim 3, wherein said electron beam is formed by acathode-anode arrangement.
 7. The system of claim 6, wherein saidCherenkov radiation is tunable by frequency.
 8. The system of claim 7,wherein said frequency is tunable by scaling the photonic crystalstructure.
 9. The system of claim 2, wherein said photonic crystalstructure comprises of no output waveguides.
 10. The system of claim 2,wherein said beam of traveling charged particles travels in an all-airchannel of said photonic crystal structure.
 11. A method of exhibitingCherenkov radiation comprising: providing a beam of charged particles;and providing a photonic crystal structure that receives said beam ofcharged particles, said charged particles moving in said photoniccrystal structure so that Cherenkov radiation is produced at allvelocities without requiring resonances in the effective materialconstants of said photonic crystal structure.
 12. The method of claim11, wherein said charged particles comprise of an electron beam.
 13. Themethod of claim 12, wherein said photonic crystal structure comprises ofoutput waveguides where said Cherenkov radiation outputs said photoniccrystal structure.
 14. The method of claim 13, wherein said Cherenkovradiation is received by one or more photomultipliers.
 15. The method ofclaim 14, wherein said dispersion of charges particles are absorbed byan absorber.
 16. The method of claim 13, wherein said electron beam isformed by a cathode-anode arrangement.
 17. The method of claim 16,wherein said Cherenkov radiation is tunable by frequency.
 18. The methodof claim 17, wherein said frequency is tunable by scaling the photoniccrystal structure.
 19. The method of claim 12, wherein said beam oftraveling charged particles travels in an all-air channel of saidphotonic crystal structure.
 20. The method of claim 12, wherein saidphotonic crystal structure comprises no output waveguides.